1. Structure of gaseous, liquid and solid bodies
The molecular kinetic theory makes it possible to understand why a substance can exist in gaseous, liquid and solid states.
Gases. In gases, the distance between atoms or molecules is on average many times more sizes the molecules themselves ( Fig.8.5). For example, at atmospheric pressure the volume of a vessel is tens of thousands of times greater than the volume of the molecules in it.
Gases are easily compressed, and the average distance between molecules decreases, but the shape of the molecule does not change ( Fig.8.6).
Molecules move at enormous speeds - hundreds of meters per second - in space. When they collide, they bounce off each other in different directions like billiard balls. The weak attractive forces of gas molecules are not able to hold them near each other. That's why gases can expand unlimitedly. They retain neither shape nor volume.
Numerous impacts of molecules on the walls of the vessel create gas pressure.
Liquids. The molecules of the liquid are located almost close to each other ( Fig.8.7), so a liquid molecule behaves differently than a gas molecule. In liquids, there is so-called short-range order, i.e., the ordered arrangement of molecules is maintained over distances equal to several molecular diameters. The molecule oscillates around its equilibrium position, colliding with neighboring molecules. Only from time to time she makes another “jump”, getting into a new equilibrium position. In this equilibrium position, the repulsive force is equal to the attractive force, i.e., the total interaction force of the molecule is zero. Time settled life water molecules, i.e., the time of its vibrations around one specific equilibrium position at room temperature, is on average 10 -11 s. The time of one oscillation is much less (10 -12 -10 -13 s). With increasing temperature, the residence time of molecules decreases.
The nature of molecular motion in liquids, first established by the Soviet physicist Ya.I. Frenkel, allows us to understand the basic properties of liquids.
Liquid molecules are located directly next to each other. As the volume decreases, the repulsive forces become very large. This explains low compressibility of liquids.
As is known, liquids are fluid, that is, they do not retain their shape. This can be explained this way. The external force does not noticeably change the number of molecular jumps per second. But jumps of molecules from one stationary position to another occur predominantly in the direction of action external force (Fig.8.8). This is why liquid flows and takes the shape of the container.
Solids. Atoms or molecules of solids, unlike atoms and molecules of liquids, vibrate around certain equilibrium positions. For this reason, solids retain not only volume, but also shape. The potential energy of interaction between solid molecules is significantly greater than their kinetic energy.
There is another important difference between liquids and solids. A liquid can be compared to a crowd of people, where individuals restlessly jostling in place, and the solid body is like a slender cohort of the same individuals who, although they do not stand at attention, maintain on average certain distances between themselves. If you connect the centers of the equilibrium positions of atoms or ions of a solid body, you get a regular spatial lattice called crystalline.
Figures 8.9 and 8.10 show the crystal lattices of table salt and diamond. The internal order in the arrangement of atoms in crystals leads to regular external geometric shapes.
Figure 8.11 shows Yakut diamonds.
In a gas, the distance l between molecules is much greater than the size of the molecules 0:" l>>r 0 .
For liquids and solids l≈r 0. The molecules of a liquid are arranged in disorder and from time to time jump from one settled position to another.
Crystalline solids have molecules (or atoms) arranged in a strictly ordered manner.
2. Ideal gas in molecular kinetic theory
The study of any field of physics always begins with the introduction of a certain model, within the framework of which further study takes place. For example, when we studied kinematics, the model of the body was a material point, etc. As you may have guessed, the model will never correspond to the actually occurring processes, but often it comes very close to this correspondence.
Molecular physics, and in particular MKT, is no exception. Many scientists have worked on the problem of describing the model since the eighteenth century: M. Lomonosov, D. Joule, R. Clausius (Fig. 1). The latter, in fact, introduced the model in 1857 ideal gas. A qualitative explanation of the basic properties of a substance based on molecular kinetic theory is not particularly difficult. However, the theory that establishes quantitative connections between experimentally measured quantities (pressure, temperature, etc.) and the properties of the molecules themselves, their number and speed of movement, is very complex. In a gas at normal pressures, the distance between the molecules is many times greater than their dimensions. In this case, the interaction forces between molecules are negligible and the kinetic energy of the molecules is much greater than the potential energy of interaction. Gas molecules can be thought of as material points or very small solid balls. Instead of real gas, between the molecules of which there are actions complex forces interaction, we will consider it The model is an ideal gas.
Ideal gas– a gas model, in which gas molecules and atoms are represented in the form of very small (vanishing sizes) elastic balls that do not interact with each other (without direct contact), but only collide (see Fig. 2).
It should be noted that rarefied hydrogen (under very low pressure) almost completely satisfies the ideal gas model.
Rice. 2.
Ideal gas is a gas in which the interaction between molecules is negligible. Naturally, when molecules of an ideal gas collide, a repulsive force acts on them. Since we can consider gas molecules, according to the model, as material points, we neglect the sizes of the molecules, considering that the volume they occupy is much less than the volume of the vessel.
Let us recall that in a physical model only those properties of a real system are taken into account, the consideration of which is absolutely necessary to explain the studied patterns of behavior of this system. No model can convey all the properties of a system. Now we have to solve a rather narrow problem: using molecular kinetic theory to calculate the pressure of an ideal gas on the walls of a vessel. For this problem, the ideal gas model turns out to be quite satisfactory. It leads to results that are confirmed by experience.
3. Gas pressure in molecular kinetic theory
Let the gas be in a closed container. Pressure gauge shows gas pressure p 0. How does this pressure arise?
Each gas molecule hitting the wall acts on it with a certain force for a short period of time. As a result of random impacts on the wall, the pressure changes rapidly over time, approximately as shown in Figure 8.12. However, the effects caused by the impacts of individual molecules are so weak that they are not registered by a pressure gauge. The pressure gauge records the time-average force acting on each unit of its surface area. sensitive element- membranes. Despite small changes in pressure, the average pressure value p 0 practically turns out to be a completely definite value, since there are a lot of impacts on the wall, and the masses of the molecules are very small.
An ideal gas is a model of a real gas. According to this model, gas molecules can be considered as material points whose interaction occurs only when they collide. When the gas molecules collide with the wall, they exert pressure on it.
4. Micro- and macroparameters of gas
Now we can begin to describe the parameters of an ideal gas. They are divided into two groups:
Ideal gas parameters
That is, microparameters describe the state of a single particle (microbody), and macroparameters describe the state of the entire portion of gas (macrobody). Let us now write down the relationship that connects some parameters with others, or the basic MKT equation:
Here: - average speed of particle movement;
Definition. – concentration gas particles – the number of particles per unit volume; ; unit - .
5. Average value of the square of the speed of molecules
To calculate the average pressure you need to know average speed molecules (more precisely, the average value of the square of the velocity). This is not a simple question. You are used to the fact that every particle has speed. The average speed of molecules depends on the movement of all particles.
Average values. From the very beginning, you need to give up trying to trace the movement of all the molecules that make up the gas. There are too many of them, and they move very difficult. We don't need to know how each molecule moves. We must find out what result the movement of all gas molecules leads to.
The nature of the movement of the entire set of gas molecules is known from experience. Molecules engage in random (thermal) motion. This means that the speed of any molecule can be either very large or very small. The direction of motion of molecules constantly changes as they collide with each other.
The speeds of individual molecules can be any, however average the value of the modulus of these speeds is quite definite. Similarly, the height of students in a class is not the same, but its average is a certain number. To find this number, you need to add up the heights of individual students and divide this sum by the number of students.
The average value of the square of the speed. In the future, we will need the average value not of the speed itself, but of the square of the speed. The average kinetic energy of molecules depends on this value. And the average kinetic energy of molecules, as we will soon see, has a very great importance throughout molecular kinetic theory.
Let us denote the velocity modules of individual gas molecules by . The average value of the square of the speed is determined by the following formula:
Where N- the number of molecules in the gas.
But the square of the modulus of any vector equal to the sum squares of its projections on the coordinate axes OX, OY, OZ. That's why
Average values of quantities can be determined using formulas similar to formula (8.9). Between the average value and the average values of the squares of projections there is the same relationship as relationship (8.10):
Indeed, equality (8.10) is valid for each molecule. Adding these equalities for individual molecules and dividing both sides of the resulting equation by the number of molecules N, we arrive at formula (8.11).
Attention! Since the directions of the three axes OH, OH And OZ due to the random movement of molecules, they are equal, the average values of the squares of the velocity projections are equal to each other:
You see, a certain pattern emerges from the chaos. Could you figure this out for yourself?
Taking into account relation (8.12), we substitute in formula (8.11) instead of and . Then for the mean square of the velocity projection we obtain:
i.e., the mean square of the velocity projection is equal to 1/3 of the mean square of the velocity itself. The 1/3 factor appears due to the three-dimensionality of space and, accordingly, the existence of three projections for any vector.
The speeds of molecules change randomly, but the average square of the speed is a well-defined value.
6. Basic equation of molecular kinetic theory
Let us proceed to the derivation of the basic equation of the molecular kinetic theory of gases. This equation establishes the dependence of gas pressure on the average kinetic energy of its molecules. After the derivation of this equation in the 19th century. and experimental proof of its validity began fast development quantitative theory, which continues to this day.
The proof of almost any statement in physics, the derivation of any equation can be done with varying degrees of rigor and convincingness: very simplified, more or less rigorous, or with the full rigor available modern science.
A rigorous derivation of the equation of the molecular kinetic theory of gases is quite complex. Therefore, we will limit ourselves to a highly simplified, schematic derivation of the equation. Despite all the simplifications, the result will be correct.
Derivation of the basic equation. Let's calculate the gas pressure on the wall CD vessel ABCD area S, perpendicular coordinate axis OX (Fig.8.13).
When a molecule hits a wall, its momentum changes: . Since the modulus of the speed of molecules upon impact does not change, then 
. According to Newton's second law, the change in the momentum of a molecule is equal to the impulse of the force acting on it from the wall of the vessel, and according to Newton's third law, the magnitude of the impulse of the force with which the molecule acts on the wall is the same. Consequently, as a result of the impact of the molecule, a force was exerted on the wall, the momentum of which is equal to .
Molecular physics made easy!
Molecular interaction forces
All molecules of a substance interact with each other through forces of attraction and repulsion.
Evidence of the interaction of molecules: the phenomenon of wetting, resistance to compression and tension, low compressibility of solids and gases, etc.
The reason for the interaction of molecules is the electromagnetic interactions of charged particles in a substance.
How to explain this?
An atom consists of a positively charged nucleus and a negatively charged electron shell. The charge of the nucleus is equal to the total charge of all the electrons, so the atom as a whole is electrically neutral.
A molecule consisting of one or more atoms is also electrically neutral.
Let's consider the interaction between molecules using the example of two stationary molecules.
Gravitational and electromagnetic forces can exist between bodies in nature.
Since the masses of molecules are extremely small, negligible forces of gravitational interaction between molecules can be ignored.
At very large distances there is also no electromagnetic interaction between molecules.
But, as the distance between molecules decreases, the molecules begin to orient themselves in such a way that their sides facing each other will have charges of different signs (in general, the molecules remain neutral), and attractive forces arise between the molecules.
With an even greater decrease in the distance between molecules, repulsive forces arise as a result of the interaction of negatively charged electron shells of the atoms of the molecules.
As a result, the molecule is acted upon by the sum of the forces of attraction and repulsion. At large distances, the force of attraction predominates (at a distance of 2-3 diameters of the molecule, the attraction is maximum), at short distances the force of repulsion prevails.
There is a distance between molecules at which the attractive forces become equal forces repulsion. This position of the molecules is called position stable equilibrium.
Molecules located at a distance from each other and connected by electromagnetic forces have potential energy.
In a stable equilibrium position, the potential energy of the molecules is minimal.
In a substance, each molecule interacts simultaneously with many neighboring molecules, which also affects the value of the minimum potential energy of the molecules.
In addition, all molecules of a substance are in continuous motion, i.e. have kinetic energy.
Thus, the structure of a substance and its properties (solid, liquid and gaseous bodies) are determined by the relationship between the minimum potential energy of interaction of molecules and the reserve of kinetic energy thermal movement molecules.
Structure and properties of solid, liquid and gaseous bodies
The structure of bodies is explained by the interaction of particles of the body and the nature of their thermal movement.
Solid
Solids have a constant shape and volume and are practically incompressible.
The minimum potential energy of interaction of molecules is greater than the kinetic energy of molecules.
Strong particle interaction.
The thermal motion of molecules in a solid is expressed only by vibrations of particles (atoms, molecules) around a stable equilibrium position.
Due to the large forces of attraction, molecules practically cannot change their position in matter, this explains the invariability of the volume and shape of solids.
Most solids have a spatially ordered arrangement of particles that form a regular crystal lattice.
Particles of matter (atoms, molecules, ions) are located at the vertices - nodes of the crystal lattice. The nodes of the crystal lattice coincide with the position of stable equilibrium of the particles.
Such solids are called crystalline.
Liquid
Liquids have a certain volume, but do not have their own shape; they take the shape of the vessel in which they are located.
Weak particle interaction.
The thermal motion of molecules in a liquid is expressed by vibrations around a stable equilibrium position within the volume provided to the molecule by its neighbors
Molecules cannot move freely throughout the entire volume of a substance, but transitions of molecules to neighboring places are possible. This explains the fluidity of the liquid and the ability to change its shape.
In liquids, molecules are quite firmly bound to each other by forces of attraction, which explains the invariance of the volume of the liquid.
In a liquid, the distance between molecules is approximately equal to the diameter of the molecule. When the distance between molecules decreases (compression of the liquid), the repulsive forces increase sharply, so liquids are incompressible.
In terms of their structure and the nature of thermal movement, liquids occupy an intermediate position between solids and gases.
Although the difference between a liquid and a gas is much greater than between a liquid and a solid. For example, during melting or crystallization, the volume of a body changes many times less than during evaporation or condensation.
Gases do not have a constant volume and occupy the entire volume of the vessel in which they are located.
The minimum potential energy of interaction between molecules is less than the kinetic energy of molecules.
Particles of matter practically do not interact.
Gases are characterized by complete disorder in the arrangement and movement of molecules.
What is the average distance between molecules of saturated water vapor at a temperature of 100°C?
Problem No. 4.1.65 from the “Collection of problems for preparing for entrance exams in physics USPTU"
Given:
\(t=100^\circ\) C, \(l-?\)
The solution of the problem:
Let's consider water vapor in some arbitrary amount equal to \(\nu\) moles. To determine the volume \(V\) occupied by a given amount of water vapor, you need to use the Clapeyron-Mendeleev equation:
In this formula, \(R\) is the universal gas constant equal to 8.31 J/(mol K). The pressure of saturated water vapor \(p\) at a temperature of 100° C is equal to 100 kPa, this known fact, and every student should know it.
To determine the number of water vapor molecules \(N\), we use the following formula:
Here \(N_A\) is Avogadro’s number, equal to 6.023·10 23 1/mol.
Then for each molecule there is a cube of volume \(V_0\), obviously determined by the formula:
\[(V_0) = \frac(V)(N)\]
\[(V_0) = \frac((\nu RT))((p\nu (N_A))) = \frac((RT))((p(N_A)))\]
Now look at the diagram for the problem. Each molecule is conditionally located in its own cube, the distance between two molecules can vary from 0 to \(2d\), where \(d\) is the length of the cube edge. The average distance \(l\) will be equal to the length of the edge of the cube \(d\):
The edge length \(d\) can be found like this:
As a result, we get the following formula:
Let's convert the temperature to the Kelvin scale and calculate the answer:
Answer: 3.72 nm.
If you do not understand the solution and you have any questions or you have found an error, then feel free to leave a comment below.
The molecules are very small; ordinary molecules cannot be seen even in the strongest light. optical microscope- but some parameters of molecules can be calculated quite accurately (mass), and some can only be very roughly estimated (dimensions, speed), and it would also be good to understand what “molecule size” is and what kind of “molecule speed” we are talking about. So, the mass of a molecule is found as “the mass of one mole” / “the number of molecules in a mole”. For example, for a water molecule m = 0.018/6·1023 = 3·10-26 kg (you can calculate more precisely - Avogadro’s number is known with good accuracy, and the molar mass of any molecule is easy to find).
Estimating the size of a molecule begins with the question of what constitutes its size. If only she were a perfectly polished cube! However, it is neither a cube nor a ball, and in general it does not have clearly defined boundaries. What to do in such cases? Let's start from afar. Let's estimate the size of a much more familiar object - a schoolchild. We have all seen schoolchildren, let’s take the mass of an average schoolchild to be 60 kg (and then we’ll see whether this choice has a significant effect on the result), the density of a schoolchild is approximately like that of water (remember that if you take a deep breath of air, and after that you can “hang” in the water, immersed almost completely, and if you exhale, you immediately begin to drown). Now you can find the volume of a schoolchild: V = 60/1000 = 0.06 cubic meters. meters. If we now assume that the student has the shape of a cube, then its size is found as the cube root of the volume, i.e. approximately 0.4 m. This is how the size turned out - less than the height (the “height” size), more than the thickness (the “depth” size). If we don’t know anything about the shape of a schoolchild’s body, then we won’t find anything better than this answer (instead of a cube we could take a ball, but the answer would be approximately the same, and calculating the diameter of a ball is more difficult than the edge of a cube). But if we have additional information (from analysis of photographs, for example), then the answer can be made much more reasonable. Let it be known that the “width” of a schoolchild is on average four times less than his height, and his “depth” is three times less. Then Н*Н/4*Н/12 = V, hence Н = 1.5 m (there is no point in making a more accurate calculation of such a poorly defined value; relying on the capabilities of a calculator in such a “calculation” is simply illiterate!). We received a completely reasonable estimate of the height of a schoolchild; if we took a mass of about 100 kg (and there are such schoolchildren!), we would get approximately 1.7 - 1.8 m - also quite reasonable.
Let us now estimate the size of a water molecule. Let's find the volume per molecule in “liquid water” - in it the molecules are most densely packed (pressed closer to each other than in the solid, “ice” state). A mole of water has a mass of 18 g and a volume of 18 cubic meters. centimeters. Then the volume per molecule is V= 18·10-6/6·1023 = 3·10-29 m3. If we do not have information about the shape of a water molecule (or if we do not want to take into account the complex shape of molecules), the easiest way is to consider it a cube and find the size exactly as we just found the size of a cubic schoolchild: d= (V)1/3 = 3·10-10 m. That's all! You can evaluate the influence of the shape of fairly complex molecules on the result of the calculation, for example, like this: calculate the size of gasoline molecules, counting the molecules as cubes - and then conduct an experiment by looking at the area of the spot from a drop of gasoline on the surface of the water. Considering the film to be a “liquid surface one molecule thick” and knowing the mass of the drop, we can compare the sizes obtained by these two methods. The result will be very instructive!
The idea used is also suitable for a completely different calculation. Let us estimate the average distance between neighboring molecules of a rarefied gas for a specific case - nitrogen at a pressure of 1 atm and a temperature of 300 K. To do this, let’s find the volume per molecule in this gas, and then everything will turn out simple. So, let’s take a mole of nitrogen under these conditions and find the volume of the portion indicated in the condition, and then divide this volume by the number of molecules: V= R·T/P·NA= 8.3·300/105·6·1023 = 4·10 -26 m3. Let us assume that the volume is divided into densely packed cubic cells, and each molecule “on average” sits in the center of its cell. Then the average distance between neighboring (closest) molecules is equal to the edge of the cubic cell: d = (V)1/3 = 3·10-9 m. It can be seen that the gas is rarefied - with such a relationship between the size of the molecule and the distance between the “neighbors” the molecules themselves occupy a rather small - approximately 1/1000 part - of the volume of the vessel. In this case, too, we carried out the calculation very approximately - there is no point in calculating such not very definite quantities as “the average distance between neighboring molecules” more accurately.
Gas laws and fundamentals of the ICT.
If the gas is sufficiently rarefied (and this is a common thing; we most often have to deal with rarefied gases), then almost any calculation is made using a formula connecting pressure P, volume V, amount of gas ν and temperature T - this is the famous “equation state of an ideal gas" P·V= ν·R·T. How to find one of these quantities if all the others are given is quite simple and understandable. But the problem can be formulated in such a way that the question will be about some other quantity - for example, about the density of a gas. So, the task: find the density of nitrogen at a temperature of 300K and a pressure of 0.2 atm. Let's solve it. Judging by the condition, the gas is quite rarefied (air consisting of 80% nitrogen and at significantly higher pressure can be considered rarefied, we breathe it freely and easily pass through it), and if this were not so, we don’t have any other formulas no – we use this favorite one. The condition does not specify the volume of any portion of gas; we will specify it ourselves. Let's take 1 cubic meter of nitrogen and find the amount of gas in this volume. Knowing the molar mass of nitrogen M = 0.028 kg/mol, we find the mass of this portion - and the problem is solved. Amount of gas ν= P·V/R·T, mass m = ν·М = М·P·V/R·T, hence density ρ= m/V = М·P/R·T = 0.028·20000/( 8.3·300) ≈ 0.2 kg/m3. The volume we chose was not included in the answer; we chose it for specificity - it’s easier to reason this way, because you don’t necessarily immediately realize that the volume can be anything, but the density will be the same. However, you can figure out that “by taking a volume, say, five times larger, we will increase the amount of gas exactly five times, therefore, no matter what volume we take, the density will be the same.” You could simply rewrite your favorite formula, substituting into it the expression for the amount of gas through the mass of a portion of gas and its molar mass: ν = m/M, then the ratio m/V = M P/R T is immediately expressed, and this is the density . It was possible to take a mole of gas and find the volume it occupies, after which the density is immediately found, because the mass of the mole is known. In general, than easier task, the more equivalent and beautiful ways to solve it...
Here is another problem where the question may seem unexpected: find the difference in air pressure at a height of 20 m and at a height of 50 m above ground level. Temperature 00C, pressure 1 atm. Solution: if we find the air density ρ under these conditions, then the pressure difference ∆P = ρ·g·∆H. We find the density in the same way as in the previous problem, the only difficulty is that air is a mixture of gases. Assuming that it consists of 80% nitrogen and 20% oxygen, we find the mass of a mole of the mixture: m = 0.8 0.028 + 0.2 0.032 ≈ 0.029 kg. The volume occupied by this mole is V= R·T/P and the density is found as the ratio of these two quantities. Then everything is clear, the answer will be approximately 35 Pa.
The gas density will have to be calculated when finding, for example, the lifting force hot air balloon of a given volume, when calculating the amount of air in scuba tanks required for breathing under water for a certain time, when calculating the number of donkeys needed to transport a given amount of mercury vapor through the desert, and in many other cases.
But the task is more complicated: an electric kettle is boiling noisily on the table, the power consumption is 1000 W, efficiency. heater 75% (the rest “goes” into the surrounding space). A stream of steam flies out of the spout - the area of the “spout” is 1 cm2. Estimate the speed of the gas in this stream. Take all the necessary data from the tables.
Solution. Let us assume that in the kettle above the water saturated steam, then a jet of saturated water vapor flies out of the spout at +1000C. The pressure of such steam is 1 atm, it is easy to find its density. Knowing the power used for evaporation Р= 0.75·Р0 = 750 W and the specific heat of vaporization (evaporation) r = 2300 kJ/kg, we will find the mass of steam formed during time τ: m= 0.75Р0·τ/r. We know the density, then it is easy to find the volume of this amount of steam. The rest is already clear - imagine this volume in the form of a column with a cross-sectional area of 1 cm2, the length of this column divided by τ will give us the speed of departure (this length takes off in a second). So, the speed of the jet leaving the spout of the kettle is V = m/(ρ S τ) = 0.75 P0 τ/(r ρ S τ) = 0.75 P0 R T/(r P M ·S) = 750·8.3·373/(2.3·106·1·105·0.018·1·10-4) ≈ 5 m/s.
(c) Zilberman A.R.
This distance can be estimated by knowing the density of the substance and the molar mass. Concentration – the number of particles per unit volume is related to density, molar mass and Avogadro's number by the relationship:
where is the density of the substance.
The reciprocal of concentration is the volume per one particle, and the distance between particles, thus, the distance between particles:
For liquids and solids, density weakly depends on temperature and pressure, therefore it is an almost constant value and approximately equal, i.e. The distance between molecules is of the order of the size of the molecules themselves.
The density of a gas is highly dependent on pressure and temperature. Under normal conditions (pressure, temperature 273 K), the air density is approximately 1 kg/m3, molar mass air is 0.029 kg/mol, then the estimate using formula (5.6) gives the value. Thus, in gases, the distance between molecules is much greater than the size of the molecules themselves.
End of work -
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From the relationship that determines the relationship between the intensity and potential of the electric field, the formula for calculating the field potential follows: where the integration is performed
Polarization of dielectrics. Free and bound charges. Main types of polarization of dielectrics
The phenomenon of the appearance of electric charges on the surface of dielectrics in an electric field is called polarization. The resulting charges are polarized
Polarization vector and electrical induction vector
For quantitative characteristics polarization of dielectrics introduce the concept of polarization vector as the total (total) dipole moment of all molecules in a unit volume of the dielectric
Electric field strength in a dielectric
In accordance with the principle of superposition, the electric field in a dielectric is vectorially composed of the external field and the field of polarization charges (Fig. 3.11).
or by absolute value
Boundary conditions for the electric field
When crossing the interface between two dielectrics with different dielectric constants ε1 and ε2 (Fig. 3.12), it is necessary to take into account the boundary forces
Electrical capacity of conductors. Capacitors
A charge q imparted to an isolated conductor creates an electric field around it, the intensity of which is proportional to the magnitude of the charge. The field potential φ, in turn, is related
According to the definition, the capacitance of the capacitor is: , where (the integral is taken along the field line between the plates of the capacitor). Hence, general formula
to calculate e
Energy of a system of stationary point charges
As we already know, the forces with which charged bodies interact are potential. Consequently, a system of charged bodies has potential energy. When the charges are removed
Current characteristics. Current strength and density. Potential drop along a current-carrying conductor
Any ordered movement of charges is called electric current. Charge carriers in conducting media can be electrons, ions, “holes” and even macroscopically
Ohm's law for a homogeneous section of a chain. Conductor resistance
There is a functional relationship between the potential drop - voltage U and the current in the conductor I, called the current-voltage characteristic of a given p For flow electric current
In a conductor, it is necessary that a potential difference be maintained at its ends. Obviously, a charged capacitor cannot be used for this purpose. Action
Branched chains. Kirchhoff's rules
An electrical circuit containing nodes is called a branched circuit. A node is a place in a circuit where three or more conductors meet (Fig. 5.14).
Resistance connection
The connection of resistances can be series, parallel and mixed. 1) Serial connection. In a series connection, the current flowing through all Moving electric charges
in a closed circuit, the current source does work. There are useful and
full time job
current source.
Interaction of conductors with current. Ampere's law
It is known that a permanent magnet exerts an effect on a current-carrying conductor (for example, a current-carrying frame); the opposite phenomenon is also known - a current-carrying conductor exerts an effect on a permanent magnet (for example
Biot-Savart-Laplace law. The principle of superposition of magnetic fields
Moving electric charges (currents) change the properties of the space surrounding them - they create a magnetic field in it. This field manifests itself in the fact that the wires placed in it
According to the Biot-Savart-Laplace law, the induction of the magnetic field created by a current element dl at a distance r from it is, where α is the angle between the current element and the radius
Moment of forces acting on a circuit with current in a magnetic field
Let us place a flat rectangular circuit (frame) with current in a uniform magnetic field with induction (Fig. 9.2).
Energy of a circuit with current in a magnetic field
A current-carrying circuit placed in a magnetic field has a reserve of energy. Indeed, in order to rotate the current-carrying circuit through a certain angle in the direction in the opposite direction its rotation in the magnetic field
Circuit with current in a non-uniform magnetic field
If the circuit with current is in a non-uniform magnetic field (Fig. 9.4), then, in addition to the torque, it is also acted upon by a force due to the presence of a magnetic field gradient. Projection of this
Work done when moving a current-carrying circuit in a magnetic field
Let's consider a piece of conductor carrying current that can move freely along two guides in an external magnetic field (Fig. 9.5). We will consider the magnetic field to be uniform and directed at an angle
Magnetic induction vector flux. Gauss's theorem in magnetostatics. Vortex nature of the magnetic field
The flow of a vector through any surface S is called the integral: , where is the projection of the vector onto the normal to the surface S at a given point (Fig. 10.1).
Fig. 10.1. TO
Magnetic field circulation theorem. Magnetic voltage Circulation of the magnetic field along closed loop
l is called the integral: , where is the projection of the vector onto the direction of the tangent to the contour line at a given point.
Relevant
Magnetic field of solenoid and toroid
Let us apply the results obtained to find the magnetic field strength on the axis of a straight long solenoid and toroid.
1) Magnetic field on the axis of a straight long solenoid.
Magnetic field in matter. Ampere's hypothesis on molecular currents. Magnetization vector
Various substances are, to varying degrees, capable of magnetization: that is, under the influence of the magnetic field in which they are placed, they acquire a magnetic moment. Some substances
When crossing the interface between two magnets with different magnetic permeabilities μ1 and μ2, the magnetic field lines experience
Magnetic moments of atoms and molecules
Atoms of all substances consist of a positively charged nucleus and negatively charged electrons moving around it. Each electron moving in orbit forms a circular current of force - h
The nature of diamagnetism. Larmore's theorem
If an atom is placed in an external magnetic field with induction (Fig. 12.1), then the electron moving in orbit will be affected by a rotational moment of forces, tending to establish the magnetic moment of the electron
Paramagnetism. Curie's law. Langevin theory
If the magnetic moment of atoms is different from zero, then the substance turns out to be paramagnetic. An external magnetic field tends to establish the magnetic moments of atoms along the
Elements of the theory of ferromagnetism. Concept of exchange forces and domain structure of ferromagnets. Curie-Weiss law
As noted earlier, ferromagnets are characterized by high degree magnetization and nonlinear dependence on. Basic magnetization curve of a ferromagnet
Forces acting on a charged particle in an electromagnetic field. Lorentz force
We already know that an Ampere force acts on a current-carrying conductor placed in a magnetic field. But the current in a conductor is the directional movement of charges. This suggests the conclusion that the force de
Motion of a charged particle in a uniform constant electric field
In this case, the Lorentz force has only an electrical component. The equation of particle motion in this case is: .
Let's consider two situations: a)
Motion of a charged particle in a uniform constant magnetic field
In this case, the Lorentz force has only a magnetic component. The equation of particle motion, written in the Cartesian coordinate system, in this case is: .
Practical applications of the Lorentz force. Hall effect
One of the well-known manifestations of the Lorentz force is the effect discovered by Hall (Hall E., 1855-1938) in 1880.
_ _ _ _ _ _
The phenomenon of electromagnetic induction. Faraday's law and Lenz's rule. Induction emf. Electronic mechanism for the occurrence of induction current in metals
Whenever the current in a conductor changes, its own magnetic field also changes. Along with it, the flux of magnetic induction that penetrates the surface covered by the conductor contour also changes.
Transient processes in electrical circuits containing inductance. Extra currents of closing and breaking
With any change in current strength in any circuit, a self-inductive emf arises in it, which causes the appearance of additional currents in this circuit, called extra currents
Magnetic field energy. Energy Density
In the experiment, the diagram of which is shown in Fig. 14.7, after the switch is opened, a decreasing current flows through the galvanometer for some time. The work of this current is equal to the work of external forces, the role of which is played by the ED
Comparison of the basic theorems of electrostatics and magnetostatics
So far we have studied static electrical and magnetic fields, that is, such fields that are created by stationary charges and constant currents.
Vortex electric field. Maxwell's first equation
The appearance of an induction current in a stationary conductor when the magnetic flux changes indicates the appearance of external forces in the circuit that set charges in motion. As we already
Maxwell's hypothesis about displacement current. Interconvertibility of electric and magnetic fields. Maxwell's third equation
Maxwell's main idea is the idea of the interconvertibility of electric and magnetic fields. Maxwell suggested that not only alternating magnetic fields are sources
Differential form of Maxwell's equations
1. Applying Stokes' theorem, we transform the left side of Maxwell's first equation to the form: .
Then the equation itself can be rewritten as, whence
Closed system of Maxwell's equations. Material equations
To close the system of Maxwell's equations, it is also necessary to indicate the connection between the vectors, and, that is, to specify the properties of the material medium in which the electron is considered
Corollaries from Maxwell's equations. Electromagnetic waves. Speed of light
Let's consider some of the main consequences that follow from Maxwell's equations given in Table 2. First of all, we note that these equations are linear. It follows that
Electric oscillatory circuit. Thomson's formula Electromagnetic vibrations
can occur in a circuit containing inductance L and capacitance C (Fig. 16.1). Such a circuit is called an oscillatory circuit. Excite to
Every real oscillatory circuit has resistance (Fig. 16.3). The energy of electrical oscillations in such a circuit is gradually spent on heating the resistance, turning into Joule heat
Forced electrical oscillations. Vector diagram method
If a source of variable EMF is included in the circuit of an electrical circuit containing capacitance, inductance and resistance (Fig. 16.5), then in it, along with its own damped oscillations,
Resonance phenomena in an oscillatory circuit. Voltage resonance and current resonance
As follows from the above formulas, at a frequency of the EMF variable ω equal to, the amplitude value of the current in the oscillatory circuit takes
Wave equation. Types and characteristics of waves
The process of propagation of vibrations in space is called a wave process or simply a wave. Waves of various natures (sound, elastic,
Electromagnetic waves
From Maxwell's equations it follows that if an alternating electric or magnetic field is excited with the help of charges, a sequence of mutual transformations will arise in the surrounding space
Energy and momentum of an electromagnetic wave. Poynting vector
The propagation of an electromagnetic wave is accompanied by a transfer of energy and momentum electromagnetic field. To verify this, let us scalarly multiply the first Maxwell equation into the differential
Elastic waves in solids. Analogy with electromagnetic waves
Laws of propagation of elastic waves in solids follow from the general equations of motion of a homogeneous elastically deformed medium: , where ρ
Standing waves
When two counterpropagating waves with the same amplitude are superimposed, standing waves arise. The appearance of standing waves occurs, for example, when waves are reflected from an obstacle. P
Doppler effect
When the source and/or receiver of sound waves move relative to the medium in which the sound propagates, the frequency ν perceived by the receiver may turn out to be about
Molecular physics and thermodynamics
Introduction. Subject and tasks of molecular physics.
Molecular physics studies the state and behavior of macroscopic objects under external influences (n
Quantity of substance
A macroscopic system must contain a number of particles comparable to Avogadro's number in order to be considered within the framework of statistical physics.
Avogadro calls the number
Gas kinetic parameters
The pressure of a gas on the wall of a container is the result of collisions of gas molecules with it. Each molecule upon collision transfers a certain impulse to the wall, therefore, it acts on the wall with n
Discrete random variable. Concept of probability
Let's look at the concept of probability using a simple example.
Let there be white and black balls mixed in a box, which are no different from each other except for color. For simplicity we will
Distribution of molecules by speed Experience shows that the velocities of gas molecules that are in an equilibrium state can have the most different meanings
– both very large and close to zero. The speed of molecules can
Basic equation of molecular kinetic theory
The average kinetic energy of translational motion of molecules is equal to: . (4.2.15) Thus, the absolute temperature is proportional to the average kinetic energy
Number of degrees of freedom of a molecule
Formula (31) determines only the energy of translational motion of the molecule. Molecules of a monatomic gas have this average kinetic energy. For polyatomic molecules, it is necessary to take into account the contribution to
Internal energy of an ideal gas
The internal energy of an ideal gas is equal to the total kinetic energy of the movement of molecules: The internal energy of one mole of an ideal gas is equal to: (4.2.20) Internal
Barometric formula. Boltzmann distribution Atmospheric pressure at height h is determined by the weight of the overlying gas layers. If the air temperature T and acceleration free fall
g do not change with altitude, then air pressure P at altitude
The first law of thermodynamics. Thermodynamic system. External and internal parameters. Thermodynamic process The word "thermodynamics" comes from the Greek words thermos - heat, and dynamics - force. Thermodynamics emerged as the science of driving forces
arising during thermal processes, about the law
Equilibrium state. Equilibrium processes
If all the parameters of the system have certain values that remain constant under constant external conditions for an indefinitely long time, then such a state of the system is called equilibrium, or
Mendeleev - Clapeyron equation
In a state of thermodynamic equilibrium, all parameters of a macroscopic system remain unchanged for as long as desired under constant external conditions. The experiment shows that for any
Internal energy of a thermodynamic system In addition to thermodynamic parameters P,V
The concept of heat capacity
According to the first law of thermodynamics, the amount of heat dQ imparted to the system goes to change its internal energy dU and the work dA that the system does on external
Lecture text
Compiled by: GumarovaSonia Faritovna The book is published in the author's edition Sub. to print 00.00.00. format 60x84 1/16.
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